A hamiltonian regularization of the burgers equation

Journal of Nonlinear Science

Volumn 16, Issue 6, 2006, Pages 615-638

  Bhat, H S ^a   Fetecau, R C ^b  

^a Columbia University ^*  (United States)

^b STANFORD UNIVERSITY   (United States)

Author keywords

Burgers equation; Leray regularization, method of characteristics; Nonlinear evolution equation; nonlocal Poisson structure.; singular limit

Indexed keywords

EID: 84867931089     PISSN: 09388974     EISSN: 14321467     Source Type: Journal    
DOI: 10.1007/s00332-005-0712-7     Document Type: Article

References (23)

1. R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, vol. 65, Academic Press, New York, 1975.
2. C. Cao, D. D. Holm, and E. S. Titi, Traveling wave solutions for a class of onedimensional nonlinear shallow water wave models, J. Dynam. Diff. Eqns. 16 (2004), 167-178. (Pubitemid 39217526)
3. H. R. Dullin, G. A. Gottwald, and D. D. Holm, Camassa-Holm, Korteweg-de Vries-5 and other asymptotically equivalent equations for shallow water waves, Fluid Dynam. Res. 33 (2003), 73-95. (Pubitemid 36785130)
4. H. R. Dullin, G. A. Gottwald, and D. D. Holm, On asymptotically equivalent shallow water wave equations, Phys. D 190 (2004), 1-14.
5. A. Degasperis, D. D. Holm, and A. N. W. Hone, Integrable and non-integrable equations with peakons, in Nonlinear Physics: Theory and Experiment, II (Gallipoli, 2002), World Sci. Publishing, River Edge, NJ, 2003, pp. 37-43.
6. G. B. Folland, Real Analysis: Modern Techniques and their Applications, Wiley-Interscience, New York, 1999.
7. J. Goodman and P. D. Lax, On dispersive difference schemes, I, Comm. Pure Appl. Math. 41(5) (1988), 591-613.
8. D. D. Holm and A. N. W. Hone, A class of equations with peakon and pulson solutions, J. Nonlinear Math. Phys. 12 (2005), no. suppl. 1, 380-394, with an appendix by H. W. Braden and J. G. Byatt-Smith.
9. H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Conservation Laws, Applied Mathematical Sciences, vol. 152, Springer-Verlag, New York, 2002.
10. D. D. Holm and M. F. Staley, Wave structure and nonlinear balances in a family of evolutionary pdes, SIAM J. Appl. Dyn. Sys. 2 (2003), 323-380.
11. A. N. W. Hone and J. P. Wang, Prolongation algebras and Hamiltonian operators for peakon equations, Inverse Problems 19 (2003), 129-145.
12. Arieh Iserles, A First Course in the Numerical Analysis of Differential Equations, Cambridge University Press, Cambridge, 1996.
13. F. John, Nonlinear Wave Equations, Formation of Singularities, University Lecture Series, vol. 2, American Mathematical Society, Providence, RI, 1990.
14. J. Leray, Essai sur le mouvement d'un fluide visqueux emplissant l'space, Acta Math. 63 (1934), 193-248.
15. C. Lattanzio and P. Marcati, Global well-posedness and relaxation limits of a model for radiating gas, J. Diff. Eqns. 190 (2003), 439-465. (Pubitemid 36579501)
16. H. Liu and E. Tadmor, Critical thresholds in a convolution model for nonlinear conservation laws, SIAM J. Math. Anal. 33 (2001), 930-945.
17. A. V. Mikhailov and V. S. Novikov, Perturbative symmetry approach, J. Phys. A 35 (2002), 4775-4790. (Pubitemid 37023811)
18. J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, second ed., Texts in Applied Mathematics, vol. 17, Springer-Verlag, New York, 1999.
19. K. Mohseni, H. Zhao, and J. E. Marsden, Shock regularization for the Burgers equation, in Proceedings of the 44th AIAA Aerospace Sciences Meeting and Exhibit (Reno, NV), January 2006, AIAA Paper 2006-1516.
20. P. J. Olver, Applications of Lie Groups to Differential Equations, second edn., Graduate Texts in Mathematics, vol. 107, Springer-Verlag, New York, 1993.
21. J. Smoller, Shock Waves and Reaction-Diffusion Equations, Grundlehren der Mathematischen Wissenschaften, vol. 258, Springer-Verlag, New York, 1983.
22. S. Schochet and E. Tadmor, The regularized Chapman-Enskog expansion for scalar conservation laws, Arch. Ratl. Mech. Anal. 119 (1992), 95-107.
23. E. Tadmor, Burgers' equation with vanishing hyper-viscosity, Comm. Math. Sci. 2 (2004), 317-324.